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Proof of the Double Bubble Conjecture in R^ N

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Proof of the Double Bubble Conjecture in R^ N PROOF OF THE DOUBLE BUBBLE CONJECTURE IN Rn BEN W. REICHARDT Abstract. The least-area hypersurface enclosing and separating two given volumes in Rn is the standard double bubble. 1. Introduction 1.1. The Double Bubble Conjecture. We extend the proof of the double bubble theorem [HMRR02] from R3 to Rn. Theorem 1.1 (Double Bubble Conjecture). The least-area hypersurface enclos- ing and separating two given volumes in Rn is the standard double soap bubble of Figure 1, consisting of three (n − 1)-dimensional spherical caps intersecting at 120 degree angles. (For the case of equal volumes, the middle cap is a flat disk.) In 1990, Foisy, Alfaro, Brock, Hodges and Zimba [FAB+93] proved the Double Bubble Conjecture in R2. In 1995, Hass, Hutchings and Schlafly [HHS95, HS00] used a computer to prove the conjecture for the case of equal volumes in R3. Arguments since have relied on the Hutchings structure theorem (Theorem 3.1), stating roughly that the only possible nonstandard minimal double bubbles are ro- tationally symmetric about an axis and consist of “trees” of annular bands wrapped around each other [Hut97]. Figures 2 and 3 show examples of “4+4” bubbles, in which the region for each volume is divided into four connected components. In 2000, Hutchings, Morgan, Ritor´eand Ros [HMRR02] used stability arguments to prove the conjecture for all cases in R3. For R3, component bounds after Hutch- ings [Hut97] guarantee that the region enclosing the larger volume is connected and the region enclosing the smaller volume has at most two components. (For equal volumes, both regions need be connected.) Eliminating “1 + 2” and “1 + 1” non- standard bubbles proved the conjecture in R3. Section 1.2 below sketches their instability argument. ([Mor00] also gives background and a proof sketch.) arXiv:0705.1601v1 [math.MG] 11 May 2007 1991 Mathematics Subject Classification. 53A10. Figure 1. The standard double bubble, consisting of three spher- ical caps meeting at 120 degree angles, is now known to be the least-area hypersurface that encloses two given volumes in Rn. 1 2 BEN W. REICHARDT Figure 2. A nonstandard minimal double bubble must be a hy- persurface of revolution about an axis L, consisting of a central bubble with layers of toroidal bands. Here we show the generating curves of a typical 4+4 double bubble, together with the associated tree T . Figure 3. The generating curves for another possible 4+4 bubble with the same tree structure as in Figure 2. In 2003, Reichardt, Heilmann, Lai and Spielman [RHLS03] extended their ar- guments to R4, and to higher dimensions Rn provided one volume is more than twice the other. In these cases, component bounds guarantee that one region is connected and the other has a finite number k of components. Eliminating 1 + k bubbles proved the conjecture in these cases. The same component bounds show that in R5 it suffices to eliminate 2 + 2 bubbles (as well as 1 + k bubbles) to prove the Double Bubble Conjecture. In R6 it suffices to eliminate also 2 + 3 bubbles. However, in Rn generally we know only that the larger region has at most three components and the smaller region has a finite number of components [HLRS99]. Here, we extend the methods of Hutchings et al. and Reichardt et al. to prove the Double Bubble Conjecture in Rn for n ≥ 3 for arbitrary volumes. We prove that j + k nonstandard bubbles are not minimizing for arbitrary finite component counts j, k. The arguments are similar in spirit to those of [RHLS03], except we take advantage of more properties of constant-mean-curvature surfaces of revolution in order to eliminate previously problematic cases. Our arguments also simplify the previous proofs in R3 and R4 because they eliminate the need for component bounds. 1.2. The Instability Argument. An area-minimizing double bubble Σ exists and has an axis of rotational symmetry L. Assume that Σ is a nonstandard double bubble. Consider small rotations about a line M orthogonal to L, chosen so that the n PROOF OF THE DOUBLE BUBBLE CONJECTURE IN R 3 Figure 4. The lines orthogonal to Σ through the points of the separating set all pass through M. Σ cannot be a minimizer. points of tangency between Σ and the rotation vectorfield v separate the bubble into at least four pieces, as in Figure 4. Then we can linearly combine the restrictions of v to each piece to obtain a vectorfield that vanishes on one piece and preserves volume. By regularity for eigenfunctions, v is tangent to certain related parts of Σ, implying that they are spheres centered on L ∩ M. In turn, this implies that there are too many spherical pieces of Σ, leading to a contradiction. This is the instability argument of [HMRR02] behind Theorem 4.2. Therefore, no such useful perturbation axis M can exist. By induction, starting at the connected components corresponding to leaves in the tree of Figure 2 we classify all possible configurations in which no such M can be found. The induc- tion ultimately shows that Σ must be a “near-graph component stack.” A global argument then finds a suitable M. Therefore, Σ cannot in fact be a minimizer. Having eliminated all nonstandard double bubbles from consideration, the only possible minimizer left is the standard double bubble. 2. Delaunay hypersurfaces Constant-mean-curvature hypersurfaces of revolution are known as Delaunay hypersurfaces [HMRR02, Hsi82, HY81, Del41, Eel87]. Let Σ ⊂ Rn be a constant- mean-curvature hypersurface invariant under the action of the group O(n) of isome- tries fixing the axis L. Σ is generated by a curve Γ in a plane containing L. Put coordinates on the plane so L is the x axis. Parameterize Γ = {x(t),y(t)} by arc- length t and let θ(t) be the angle from the positive x-direction up to the tangent to Γ. Then Γ is determined by the differential equations x˙ = cos θ y˙ = sin θ (1) cos θ θ˙ = −(n − 1)H + (n − 2) y 4 BEN W. REICHARDT Here, tan θ =y/ ˙ x˙ is the slope of Γ, κ = −θ˙ is the planar curvature of Γ with respect to the normal N = (sin θ, − cos θ), and H is the (constant) mean curvature of Σ with respect to N (an average of the planar curvature κ and the curvature due to rotating about L). The force of Γ with respect to N is a constant given by (2) F = yn−2(cos θ − Hy) . Note that from Eqs. (1) and (2), (n − 1)(n − 2) (3) θ¨ = − F sin θ . yn Theorem 2.1 ([HMRR02, Prop. 4.3]). Let Γ be a complete upper half-planar gen- erating curve that, when rotated about L, generates a hypersurface Σ with constant mean curvature. The pair (H, F ) determines Γ up to horizontal translation. (1) If H = 0 and F 6= 0, then Γ is a curve of catenary type and Σ is a hypersurface of catenoid type. (2) If HF < 0, then Γ is a locally convex curve and Σ is a nodoid. (3) If HF > 0, then Γ is a periodic graph over L and Σ is an unduloid or a cylinder. (4) If H = F =0, then Γ is a ray orthogonal to L and Σ is a vertical hyperplane. (5) If F =0 and H 6=0, then Γ is a semi-circle and Σ is a sphere. See Figure 5. The Delaunay hypersurfaces with nonzero mean curvature are the sphere, unduloid and nodoid. If Σ has positive mean curvature upward then it must be a nodoid. If Γ is not graph, then Σ must be either a nodoid or a hyperplane. Lemma 2.2 (Force balancing [HMRR02, Lemma 4.5]). Assume that three gener- ating curves Γi, i =0, 1, 2, of Delaunay hypersurfaces meet at a point p. Consider normals turning clockwise about p. If the curvatures with respect to these normals satisfy H0 + H1 + H2 =0, then the forces with respect to these normals satisfy (4) F0 + F1 + F2 =0 . The lemma follows from Eq. (2). 3. Structure of minimal double bubbles A double bubble is a piecewise-smooth oriented hypersurface Σ ⊂ Rn consisting of three compact pieces Σ1, Σ2 and Σ0 (smooth on their interiors), with a common boundary such that Σ1 ∪ Σ0, Σ2 ∪ Σ0 enclose two regions R1, R2, respectively, of given volumes. The work of Almgren [Alm76] (see [Mor00, Ch. 13]) and Hutchings establishes the existence and structure. Theorem 3.1 ([Hut97, Theorem 5.1]). Any nonstandard minimal double bubble is a hypersurface of revolution about some line L, composed of pieces of constant- mean-curvature hypersurfaces meeting in threes at 120 degree angles. The bubble is a topological sphere with a finite tree T of annular bands attached, as in Figure 2. The two caps of the bottom component are pieces of spheres, and the root of the tree has just one branch. Hence, any minimal double bubble is determined by an upper half-planar dia- gram of generating curves that, when rotated about L, generate the double bubble. n PROOF OF THE DOUBLE BUBBLE CONJECTURE IN R 5 Figure 5. Smooth regions of the cluster are parts of “Delaunay” hypersurfaces of revolution. Plotted are the generating curves of Delaunay hypersurfaces in R3 all starting at (x, y) = (0, 1) and θ = 0, with increasing mean curvature from top to bottom (positive downward at x = 0): three unduloids, a sphere, then three nodoids. Below are examples of a catenoid and a vertical hyperplane, the Delaunay hypersurfaces of zero mean curvature.
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